Bigeleisen equation

The one-frequency model represented by Eqs. (11.3)-(11.8) shows single isotopic frequency expressions for the MMI (mass/moment of inertia), ZPE (vibrational zero-point energy), and EXC (excited vibrations) terms of the usual Bigeleisen equation [21]. The extra term tun is the truncated Bell tunnel correction [22], used here to provide a simple way to express a tunneling effect in terms of a reaction-coordinate frequency, vh... 

The fractionation factor, which is the equilibrium constant Kf = [C][D]/[A][B], was calculated using the Bigeleisen equation as Kp = 1.11 at 25°C (Shiner and Hartshorn, 1972). This reaction might serve as a model for a maximum expected a-deuterium isotope effect in which a CH2=CHD group is changed to a cyclopropyl-D group. An experimental example is the equilibrium isotope effect K — 1.109 at 29°C in l,5-dimethyl-2-deuteriosemibullvalene [2] (Askani et al., 1982)... 

Disregarding the tunnel effects (see Sect. 1.5) and staying within the approximation rigid rotator-harmonic oscillator, one may, for the biomolecular reaction A -h B), calculate the kinetic isotopic effect (ratio between the reaction rate constant of the compound with the light isotope and the rate constant K2 of the compound containing the heavy isotope) from the Bigeleisen equation ... 

Calculation of the maximum isotope effect for C-S bond breakage is simple. Some of the C-S bond stretching frequencies of the relevant sulfonium salts can be found in literature, but some can not. In general, the stretching frequencies of C-S bonds of various sulfur-containing compounds lie between 700 and 800 cm The Bigeleisen equation for calculation of the kinetic isotope effect appears complicated ... 

Bigeleisen and Mayer (Historical Vignettes 4.1 and 4.2), recognizing that the term in Equation 4.78 involving isotopic masses would always cancel in the calculation of... 

Considerations like the above led the German statistical mechanician L. Waldman independently to an equation similar to the (si/s2)f equation of Bigeleisen and Mayer. The foregoing can be regarded as an independent proof of the Teller-Redlich product rule but this statement depends on the assumption of no rotational-vibrational interaction. 

The use of reduced isotopic partition function ratios to study kinetic isotope effects was first undertaken by Bigeleisen this work was corrected and elaborated by Bigeleisen and Wolfsberg. References are cited at the end of this chapter. Application of the equations developed above to specific chemical reactions will be found in Chapter 10, where other theoretical approaches will also be presented. 

Using a sum-of-squares rule from theoretical vibrational spectroscopy, Bigeleisen and Mayer (1947) then showed that, under the conditions relevant to Equation (4),... 

Mossbauer spectroscopy involves the measurement of minute frequency shifts in the resonant gamma-ray absorption cross-section of a target nucleus (most commonly Fe occasionally Sn, Au, and a few others) embedded in a solid material. Because Mossbauer spectroscopy directly probes the chemical properties of the target nucleus, it is ideally suited to studies of complex materials and Fe-poor solid solutions. Mossbauer studies are commonly used to infer properties like oxidation states and coordination number at the site occupied by the target atom (Flawthome 1988). Mossbauer-based fractionation models are based on an extension of Equations (4) and (5) (Bigeleisen and Mayer 1947), which relate a to either sums of squares of vibrational frequencies or a sum of force constants. In the Polyakov (1997)... 

Bigeleisen J (1949) The relative velocities of isotopic molecules. J Chem Phys 17 675-678 Bigeleisen J (1955) Statistical mechanics of isotopic systems with small quantum corrections. I. General considerations and the rule of the geometric mean. J Chem Phys 23 2264-2267 Bigeleisen J (1998) Second-order correction to the Bigeleisen-Mayer equation due to the nuclear field shift. Proc National Acad Sci 95 4808-4809... 

Based on equation 11.41, the difference between the Helmholtz free energies of formation of two isotopic molecules with respect to their gaseous atoms depends on the shift of vibrational frequencies between heavy and light isotope-bearing compounds—i.e., according to Bigeleisen and Mayer (1947),... 

Table 11.4 Separative effect / and isotopic fractionation constant K for heavy isotopes, computed through equation 11.47. is angular totally symmetric stretching frequency derived from Raman spectra (see Bigeleisen and Mayer, 1947 for references).

The theory of IEs was formulated by Bigeleisen and Mayer.9 The IE on the acid-base reaction of Equation (1) is defined as the ratio of its acidity constant KA to the acidity constant of the isotopic reaction, Equation (2). The ratio KJ KA is then the equilibrium constant XEIE for the exchange reaction of Equation (3). That equilibrium constant may be expressed in terms of the partition function Q of each of the species, as given in Equation (4), which ignores symmetry numbers. 

The expected characteristics of 7 can best be determined by an examination of the original derivation of the Y equation. Bigeleisen and Mayer (7) derived the equation... 

In the G(w)-approximation, introduced by Bigeleisen and Mayer (8) and later extended to higher orders by Bigeleisen (4), the reduced partition function ratio (Equation 1) is expanded in terms of the isotope frequency shifts, Au, = — Ui. The first three terms are... 

Subdivision of the Range of the Expansion Variable. In substituting Equation 42 into the infinite series expression for In b u). Equation 32, Bigeleisen and Ishida used a constant range,... 

The two parameters y and 8, which characterize a Jacobi polynomial, were varied, and RMSE values, as defined by Equations 58, 59, and 60, were numerically evaluated using an IBM-360 computer. Preliminary tests showed that satisfactory integrations were achieved by summing over 50 equally spaced points. The RMSE-surface was mapped for both the Bigeleisen-Ishida formula. Equation 44, and the modified one. Equation 52. Naturally, the best polynomial may depend on the order and... 

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